I recently needed to add a second set of tags to a website. The easiest way to do this is add a new taxonomy. There’s a plugin that will do this, but the tidiest method I found is to just add a code snippet to functions.php. This code will allow the addition of a new set of tags called New Tags:

/* Create taxonomy */
 add_action( 'init', 'build_taxonomies', 0 );
 function build_taxonomies() {
  'hierarchical' => false,
  'label' => 'New Tags',
  'query_var' => true,
  'rewrite' => true

If you set ‘hierarchical’ to true, the new taxonomy will mirror Categories rather than Tags.
Showing the new taxonomy in posts with a link needs another snippet of code to be added to the appropriate template(s):

<a href="<?php echo get_tag_link( $blogger->term_id ) ?>"
<?php echo get_the_term_list( $post->ID,'newtags', 'beforeterm ',' afterterm' ); ?>

I found that I had to reset permalinks before the new taxonomy would work.

You can customise the way that the new taxonomy is displayed by creating a new template called taxonomy-newtags.php.